Thursday, 4 May 2017

Binary Logic



Classification of persons in Binary Logic:

1. Truth Teller: Someone who always tells the truth (all statements made has to be assumed to be true).

2. Liar: Someone who always lies (all statements made has to be assumed to be false).

3. Alternator: Someone who alternately lies and tells the truth in any order (any two statements in a row will consist of exactly one true and one false statement).

Some typical statements and their implications:

i) I am a liar – no liar and truth teller will ever make this statement as it would end up becoming a true and false statement respectively. Hence if such a statement is given, we can conclude that the speaker is definitely an Alternator and the given statement is false. Also statements immediately preceding and following it from the same speaker will be true statements.

ii) I am not a truth teller – like previous example this statement too can only be made by an Alternator and the nature of the statement shall be true. Also statements immediately preceding and following it from the same speaker will be false statements.

iii) I am an alternator – this can either be a true statement of an Alternator or a false statement of a Liar. In both cases the immediately preceding and following statements of the same speaker will be false.

iv) I am not an alternator – this can be a false statement of an Alternator or a true statement of a Truth Teller. In both cases the immediately preceding and following statements of the same speaker will be true.

In case one of the above types of statements is given, then one can start with them and work out few definitely correct/false statements which shall allow solving the set (eg #1). In the absence of such direct clues, one has to work out different possibilities to get to the answer (eg #2).

#1. Each of three friends A, B and C are fond of exactly one of the three fruits – apple, banana and orange with no two of them liking the same fruit. Also it is known that each of them can be any of truth tellers, liars and alternators. On being asked about their choice of fruits and talking patterns they gave the following answers –

A:
I like apple

B does not like banana

C is an alternator


B:
I like orange

A is a liar

C likes apple


C:
I like banana

I am not an alternator

B likes apple

The person who likes apple is a:

a) Truth Teller     b) Liar     c) Alternator     d) Either b or c

Sol. After browsing through all statements made by A, B and C one can make out that C’s second statement is either a true statement of a truth teller or a false statement of an alternator – irrespective of which C’s first and third statement have to be true. This gives us the following result –

A
Orange
B
Apple
C
Banana

Based on above we can determine that A’s first statement is false and the second one is true. As the three of them have to be one of truth tellers/liars/alternators; we can also conclude that A is an alternator and her third statement is hence false (which means C is not an alternator). This leads us to the conclusion that C is a truth teller; thus C’s second statement also can be marked as true.

Hereafter when we analyse B’s statements we realise they all are false; so B must be a liar. With ‘T’ standing for True and ‘F’ for False the nature of statements will be as follows:

A:
I like apple
F

B does not like banana
T

C is an alternator
F



B:
I like orange
F

A is a liar
F

C likes apple
F



C:
I like banana
T

I am not an alternator
T

B likes apple
T

Ans. Option b


#2. There is exactly one truth teller, one liar and one alternator among A, B and C. When asked about their identities they gave the following responses –

A:
I am a truth teller

C is an alternator

B is a liar


B:
I am not a liar

C is an alternator

A is a liar


C:
I am a truth teller

B is not a liar

A is not a truth teller

Given that there are only two males among them - the truth teller and liar; who is the only female among the three?

a) A     b) B     c) C     d) Anyone of them
 
Sol. Here as none of the statements can be termed as definitely true or false just by reading them, we can start by assuming each one of them as truth teller respectively.

Let A be the truth teller in case 1, B in case 2 and C in case 3; then we get the following result –



Case 1
Case 2
Case 3
A:
I am a truth teller
T
F
F

C is an alternator
T
T
F

B is a liar
T

F





B:
I am not a liar
F
T
T

C is an alternator
T
T
F

A is a liar

T
T





C:
I am a truth teller


T

B is not a liar


T

A is not a truth teller


T

In case 1 as A’s all statements are true, B is supposed to be a liar. However B’s second statement concurs with that of A’s and becomes true. As a liar cannot make a true statement, case 1 is not correct.

Similarly case 2 can also be eliminated as B is assumed to be a truth teller and his third statement terms A as a liar. However they both make same second statements and the assumed liar (A) ends up making a correct statement.

With case 1 & 2 eliminated we can conclude case 3 has to be correct which means C is the truth teller. As per C, B is not a liar, hence B must be an alternator and also A must be the liar.
As the female has to be the alternator here, B is the only female among the three.

Ans. Option b

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